Methods and systems for solving a problem on a quantum computer

Abstract

A method of solving a problem can include providing a fermionic Hamiltonian, transformation of the fermionic Hamiltonian to qubit operators, transformation of the fermionic Hamiltonian in qubit operators to a mean-field Hamiltonian, and embedding the Hamiltonian onto a quantum computer. Such systems and methods may improve upon existing methods for solving electronic structure problems on a computer by adapting the problem to available hardware, reducing computational cost, and reducing the number of required qubits to solve electronic structure problems for larger number of atoms.

What is claimed is: 
     
         1 . A method of solving a problem, using a quantum computer operably coupled with a classical computing system, comprising actions of:
 providing, at the classical computing system, a provided Hamiltonian, wherein an expectation value of the provided Hamiltonian provides a variational upper bound for a target eigenvalue thereof;   generating, at the classical computing system, a set of entanglers using the provided Hamiltonian;   selecting, at the classical computing system, a subset of entanglers from the set;   determining, at the quantum computer, corresponding amplitudes of the selected entanglers, as a first iteration of the method;   repeating the action of determining, in a first iteration, with the determined amplitudes of the selected entanglers, until a first stopping condition has been met;   wherein the provided Hamiltonian is a qubit Hamiltonian.   
     
     
         2 . The method of  claim 1 , wherein the action of determining comprises an action of, if the first stopping condition has been met, obtaining an expectation value of the provided Hamiltonian based on the selected entanglers and the determined amplitude obtained in a last repetition of the action of determining in the first iteration, wherein the expectation value gives an estimate of the target eigenvalue of the provided Hamiltonian as a solution to the problem. 
     
     
         3 . The method of  claim 2 , wherein the first stopping condition is one of:
 reaching a threshold change of the expectation value of the provided Hamiltonian after an instance of the action of determining;   performing a number of first iterations;   evaluating a pre-set number of entanglers; and   achieving a pre-set threshold for the expectation value of the provided Hamiltonian.   
     
     
         4 . The method of  claim 1 , further comprising actions of:
 dressing, at the classical computing system, the provided Hamiltonian to obtain a transformed Hamiltonian, and;   restarting the actions of: providing, generating, selecting, determining, repeating, and dressing, as a second iteration, using the transformed Hamiltonian as the provided Hamiltonian, until a second stopping condition has been met.   
     
     
         5 . The method of  claim 4 , wherein the action of restarting comprises an action of, if the second stopping condition has been met, calculating a second expectation value of the provided Hamiltonian based on the transformed Hamiltonian obtained in a last repetition of the action of dressing in the second iteration, wherein the second expectation value gives an estimate of the target eigenvalue of the provided Hamiltonian as a solution to the problem. 
     
     
         6 . The method of  claim 5 , wherein the second stopping condition is one of:
 reaching a threshold change of the expectation value of the provided Hamiltonian between second iterations;   performing a number of second iterations;   evaluating a pre-set number of entanglers; and   achieving a pre-set threshold for the expectation value of the provided Hamiltonian.   
     
     
         7 . The method of  claim 1 , wherein at least one coordinate of the qubit Hamiltonian is parameterized by a spin coherent state. 
     
     
         8 . The method of  claim 7 , wherein the spin coherent state comprises an expression in spherical polar coordinates on a Bloch sphere. 
     
     
         9 . The method of  claim 1 , wherein the qubit Hamiltonian is in a form of a linear equation comprising Pauli words. 
     
     
         10 . The method of  claim 1 , wherein the qubit Hamiltonian is parameterized in Pauli Z rotations. 
     
     
         11 . The method of  claim 10 , wherein the qubit Hamiltonian parameterized in Pauli Z rotations comprises an Ising-type Hamiltonian. 
     
     
         12 . The method of  claim 1 , wherein the action of providing comprises an action of transforming a fermionic Hamiltonian into a qubit Hamiltonian. 
     
     
         13 . The method of  claim 12 , wherein the action of transforming comprises performing one of: a Jordan-Wigner transformation, the Bravyi-Kitaev method, and the Parity method. 
     
     
         14 . The method of  claim 1 , wherein the provided Hamiltonian is constrained with respect to an operator and an eigenvalue of the operator. 
     
     
         15 . The method of  claim 14 , wherein the operator comprises a commutation relation with the provided Hamiltonian. 
     
     
         16 . The method of  claim 14  wherein the operator is at least one of:
 a number operator; and 
 a spin operator, wherein the spin operator is at least one of:
 a total spin-squared operator, and 
 a projection of a total spin operator. 
 
 
     
     
         17 . The method of  claim 1 , wherein the selected entanglers are represented, on the quantum computer, as multi-qubit entanglement gates in a quantum circuit. 
     
     
         18 . The method of  claim 1 , wherein the selected entanglers comprise at least one Pauli entangler, which takes a form of a Pauli word. 
     
     
         19 . The method of  claim 1 , wherein the action of selecting comprises selecting entanglers from a Direct Interaction Set (DIS). 
     
     
         20 . The method of  claim 19 , wherein the action of selecting comprises an action of calculating derivatives for each entangler in the set. 
     
     
         21 . The method of  claim 20 , wherein the derivative comprises a first order derivative. 
     
     
         22 . The method of  claim 21 , wherein at least one entangler that has a non-zero value in the first order derivative is selected. 
     
     
         23 . The method of  claim 21 , wherein the derivative comprises a second order derivative. 
     
     
         24 . The method of  claim 21 , wherein at least one entangler that has a close-to-zero value in the first order derivative and has a negative value in the second order derivative is selected. 
     
     
         25 . The method of  claim 1 , wherein the selected entanglers reduce the expectation value of the provided Hamiltonian. 
     
     
         26 . The method of  claim 1 , wherein at least one of the selected entanglers is factorized into at least three entanglers, each of which involves a reduced number of qubits. 
     
     
         27 . The method of  claim 4 , wherein the action of dressing uses the selected entanglers, and the determined amplitude. 
     
     
         28 . The method of  claim 1 , wherein the eigenvalue is a ground-state energy. 
     
     
         29 . The method of  claim 1 , wherein the quantum computer comprises one of: a universal quantum computer, and a quantum annealer. 
     
     
         30 . A quantum computer comprising:
 a plurality of qubits;   a qubit Hamiltonian; and   a quantum circuit comprising a set of quantum logic gates to perform gate operations;   wherein the quantum logic gates are constructed based at least on a set of entanglers selected from a Direct Interaction Set (DIS).